Editing Spectral theorem (section)

===Spectral subspaces and projection-valued measures===

In the absence of (true) eigenvectors, one can look for a "spectral subspace" consisting of an ''almost eigenvector'', i.e, a closed subspace <math>V_E</math> of <math>V</math> associated with a [[Borel set]] <math>E \subset \sigma(A)</math> in the [[Spectrum_(functional_analysis)|spectrum]] of <math>A</math>. This subspace can be thought of as the closed span of generalized eigenvectors for <math>A</math> with eigen''values'' in <math>E</math>.<ref>{{harvnb|Hall|2013}} Theorem 7.2.1</ref> In the above example, where <math> [A f](t) = t f(t), \;</math> we might consider the subspace of functions supported on a small interval <math>[a,a+\varepsilon]</math> inside <math>[0,1]</math>. This space is invariant under <math>A</math> and for any <math>f</math> in this subspace, <math>Af</math> is very close to <math>af</math>. Each subspace, in turn, is encoded by the associated projection operator, and the collection of all the subspaces is then represented by a [[projection-valued measure]]. 

One formulation of the spectral theorem expresses the operator {{math|''A''}} as an integral of the coordinate function over the operator's spectrum <math>\sigma(A)</math> with respect to a projection-valued measure.<ref>{{harvnb|Hall|2013}} Theorem 7.12</ref>
<math display="block"> A = \int_{\sigma(A)} \lambda \, d \pi (\lambda).</math>When the self-adjoint operator in question is [[compact operator|compact]], this version of the spectral theorem reduces to something similar to the finite-dimensional spectral theorem above, except that the operator is expressed as a finite or countably infinite linear combination of projections, that is, the measure consists only of atoms.